On the Dynamics of the Error Floor Behavior in (Regular) LDPC ... - arXiv

SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY IN FEBRUARY, 2009

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On the Dynamics of the Error Floor Behavior in (Regular) LDPC Codes

arXiv:0902.1786v2 [cs.IT] 11 Feb 2009

Christian Schlegel, Senior Member, IEEE, Shuai Zhang

Abstract—It is shown that dominant trapping sets of regular LDPC codes, so called absorption sets, undergo a two-phased dynamic behavior in the iterative message-passing decoding algorithm. Using a linear dynamic model for the iteration behavior of these sets, it is shown that they undergo an initial geometric growth phase which stabilizes in a final bit-flipping behavior where the algorithm reaches a fixed point. This analysis is shown to lead to very accurate numerical calculations of the error floor bit error rates down to error rates that are inaccessible by simulation. The topology of the dominant absorption sets of an example code, the IEEE 802.3an (2048, 1723) regular LDPC code, are identified and tabulated using topological relationships in combination with search algorithms. Index Terms—absorption sets, error floor, Low-Density ParityCheck codes.

I. S UMMARY

T

HE error floor in modern graph-based error control codes such as low-density parity-check codes is caused by inherent structural weaknesses in the code’s interconnect network. The iterative message passing algorithm cannot overcome these weaknesses and gets trapped in error patterns which are easily identifiable as erroneous (in LDPC codes), and are thus not valid codewords, but difficult to overcome or correct [1], [2]. These weaknesses were termed trapping sets by Richardson in [3], a summary definition for the patterns on which the message passing algorithm fails for Gaussian channels. These trapping sets are dependent on the code, the channel used, and to a lesser degree also on the details of the decoding algorithm. Prior work in identifying the weaknesses of LDPC codes on erasure channels led to the definition of stopping sets in [4]. Stopping sets, being the weaknesses of LDPC codes on erasure channels, also play a role on Gaussian channels, but are not typically the dominant error mechanisms. In [5] the authors define absorption sets, which are the subgraphs of the code graph on which the Gallager bit-flipping decoding algorithms fail for binary symmetric channels. The authors observed that these absorption sets also show up as the dominant trapping sets in certain structured LDPC codes. In [6] they devise post-processing methods to reduce the effects of these absorption sets and lower the error floor of the codes in question. C. Schlegel and S. Zhang are with the High Capacity Digital Communications Laboratory (HCDC), Electrical and Computer Engineering Department, University of Alberta, Edmonton AB, T6G 2V4, CANADA (e-mail: {schlegel, szhang4}@ece.ualberta.ca).

In this paper we present a linear algebraic approach to the dynamic behavior of absorption sets. We show that these sets follow a geometric growth phase during early iterations where messages inside the absorption set grow towards a largest eigenvector which characterizes the absorption set. The seemingly erratic behavior of the messages at early iterations is due to the decreasing influence of lesser eigenvectors. We define the gain of an absorption set and show how it affects the influence of the extrinsic messages that flow into the absorption set at each iteration from the remainder of the code network. The importance of set extrinsic information was already informally observed in [7], who reported a lowering of the error floor with increased extrinsic connectivity. We use our analysis to produce accurate error formulas for the error floor BER/FER and support these results with importance sampling simulations targeting the absorption sets. As illustration we carefully identify and classify absorption sets of the regular (2048, 1723) LDPC code recently designed in [8], which is used in the IEEE 802.3an standard. Topological features of dominant absorption sets are identified and a search algorithm is presented which finds the leading dominant sets. II. BACKGROUND Stopping sets completely determine the performance of graph-based decoding of LDPC codes on erasure channels, i.e., on channels where the transmitted binary symbols are either received correctly, or are erased. A complete statistical treatment of stopping sets was given in [4]. Aptly named, a stopping set is a subset of uncorrected variable nodes where the decoder stops, i.e., makes no further correction progress. It is simply defined as: Definition 1. A stopping set S is a set of variable nodes, all of whose neighboring check nodes are connected to the set S at least twice. Fig.1 shows an example of a stopping set. It is quite straightforward to see that if erasure decoding is performed following Gallager’s decoding algorithm [9] the variable values in the stopping set cannot be reconstructed. Valid codewords are trivially stopping sets, but the set of stopping sets is larger than the set of valid codewords. An absorption set is an extension of the notion of a stopping set to the binary-symmetric channels [5], [6], and is defined as:

c 2009 IEEE 0000–0000/00$00.00

SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY IN FEBRUARY, 2009

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